Currently, along with the application of IC card being continuously increasing and becoming more and more popular, many applications are emerging, in which IC cards are used for small amount offline payment, such as an e-wallet system. The typical transaction process of said system is as follows: a terminal sends a initializing command to a card in order to initiate a transaction; the initializing command is processed by an IC card to check whether or not the key index number contained in the initializing command is supported and whether or not there is sufficient balance; a pseudo random number and a Procedure Key (SESPK) is then generated to verify the first Message Authorization code (MAC1) and send it to the terminal as a response; the first Message Authorization code (MAC1) is generated for the pseudo random number by a Secure Access Module (PSAM) of the terminal; Thereafter, the validity of the first Message Authorization code (MAC1) is verified by the IC card, and the consumed amount will be subtracted from the balance of the e-wallet, and a second Message Authorization code (MAC2) will be generated to enable the Secure Access Module to check its validity and generate a Transaction Certificate (TC) to be verified by a host. After the second Message Authorization code (MAC2) generated by the IC card is successfully verified by the Secure Access Module (PSAM), the transaction will be authorized by the terminal. However, the existing e-wallet system employs a symmetric key system, which has the following deficiencies: 1. the transfer process of the key is complex due to the use of the symmetric keys because the keys are confidential and sensitive data, and thus it requires a secure and reliable mechanism for the transfer process so as to ensure the secure transfer of it, meanwhile, the corresponding management mechanism is also very complex; 2. there are flaws in the security of the secure authentication module: in the symmetric key system, a secure authentication module (such as SAM card) need to be deployed in the terminal in order to verify the validity of an IC card. In reality, however, the terminal users are loosely managed by the card issuers, and thus it is difficult to ensure the security of the use of the secure authentication module (such as SAM card), therefore, there is a potential security risk as follows: an attacker will perform illegal attacks on the secure authentication module, such as the SAM card; 3. it is difficult to update the keys for the following reasons: since the keys used for checking the validity of the IC card are stored in the Secure Access Module (such as SAM card), and thus if a new keys need to be used, the keys in the Secure Access Module (such as SAM card) should be updated, however, generally, the procedure of updating the keys in the Secure Access Module (such as SAM card) is achieved by means of ciphertext plus certification, thereby the procedure is very complex, and thus it is difficult to realize it. Moreover, in terms of transaction functions, the e-wallet system just provides a solution for small amount offline payment, and fails to provide the function for large amount online transaction.